Groups of prime generalized exponent

Abstract

A group G satisfies a positive generalized identity of degree n if there exist elements g1gn G such that xgn=1 for all x G. The minimum degree of such an identity is called the generalized exponent of G. Among other things, we prove that every finitely generated solvable group satisfying a positive generalized identity of prime degree is a finite p-group. Consequently, we show that every finite group with a positive generalized identity of degree 5 is a 5-group of exponent dividing 25. © 2017 World Scientific Publishing Company.